Integrand size = 41, antiderivative size = 139 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\frac {x \sqrt {d+e x^2}}{2 c}+\frac {(5 c d-2 b e) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}}-\frac {(2 c d-b e)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {e} \sqrt {c d-b e}} \]
1/2*(-2*b*e+5*c*d)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c^2/e^(1/2)-(-b*e+2* c*d)^(3/2)*arctanh(x*e^(1/2)*(-b*e+2*c*d)^(1/2)/(-b*e+c*d)^(1/2)/(e*x^2+d) ^(1/2))/c^2/e^(1/2)/(-b*e+c*d)^(1/2)+1/2*x*(e*x^2+d)^(1/2)/c
Time = 0.39 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {-c x \sqrt {d+e x^2}+\frac {2 (2 c d-b e) \sqrt {2 c^2 d^2-3 b c d e+b^2 e^2} \text {arctanh}\left (\frac {-b e+c \left (d-e x^2+\sqrt {e} x \sqrt {d+e x^2}\right )}{\sqrt {2 c^2 d^2-3 b c d e+b^2 e^2}}\right )}{\sqrt {e} (c d-b e)}+\frac {(5 c d-2 b e) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{\sqrt {e}}}{2 c^2} \]
-1/2*(-(c*x*Sqrt[d + e*x^2]) + (2*(2*c*d - b*e)*Sqrt[2*c^2*d^2 - 3*b*c*d*e + b^2*e^2]*ArcTanh[(-(b*e) + c*(d - e*x^2 + Sqrt[e]*x*Sqrt[d + e*x^2]))/S qrt[2*c^2*d^2 - 3*b*c*d*e + b^2*e^2]])/(Sqrt[e]*(c*d - b*e)) + ((5*c*d - 2 *b*e)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]])/Sqrt[e])/c^2
Time = 0.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {1387, 318, 25, 27, 398, 224, 219, 291, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{b d e+b e^2 x^2-c d^2+c e^2 x^4} \, dx\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\frac {b d e-c d^2}{d}+c e x^2}dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\int -\frac {e \left (e (5 c d-2 b e) x^2+d (3 c d-b e)\right )}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{2 c e}+\frac {x \sqrt {d+e x^2}}{2 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {d+e x^2}}{2 c}-\frac {\int \frac {e \left (e (5 c d-2 b e) x^2+d (3 c d-b e)\right )}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{2 c e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {d+e x^2}}{2 c}-\frac {\int \frac {e (5 c d-2 b e) x^2+d (3 c d-b e)}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{2 c}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {x \sqrt {d+e x^2}}{2 c}-\frac {\frac {2 (2 c d-b e)^2 \int \frac {1}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{c}-\frac {(5 c d-2 b e) \int \frac {1}{\sqrt {e x^2+d}}dx}{c}}{2 c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {x \sqrt {d+e x^2}}{2 c}-\frac {\frac {2 (2 c d-b e)^2 \int \frac {1}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{c}-\frac {(5 c d-2 b e) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{c}}{2 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x \sqrt {d+e x^2}}{2 c}-\frac {\frac {2 (2 c d-b e)^2 \int \frac {1}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{c}-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) (5 c d-2 b e)}{c \sqrt {e}}}{2 c}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {x \sqrt {d+e x^2}}{2 c}-\frac {\frac {2 (2 c d-b e)^2 \int \frac {1}{-\frac {(c d e+(c d-b e) e) x^2}{e x^2+d}+c d-b e}d\frac {x}{\sqrt {e x^2+d}}}{c}-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) (5 c d-2 b e)}{c \sqrt {e}}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x \sqrt {d+e x^2}}{2 c}-\frac {\frac {2 (2 c d-b e)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{c \sqrt {e} \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) (5 c d-2 b e)}{c \sqrt {e}}}{2 c}\) |
(x*Sqrt[d + e*x^2])/(2*c) - (-(((5*c*d - 2*b*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c*Sqrt[e])) + (2*(2*c*d - b*e)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[2* c*d - b*e]*x)/(Sqrt[c*d - b*e]*Sqrt[d + e*x^2])])/(c*Sqrt[e]*Sqrt[c*d - b* e]))/(2*c)
3.3.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* (x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 1.00 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {2 \left (b e -2 c d \right )^{2} \operatorname {arctanh}\left (\frac {\left (b e -c d \right ) \sqrt {e \,x^{2}+d}}{x \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}\right )}{\sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}-\frac {\sqrt {e \,x^{2}+d}\, c x \sqrt {e}-2 \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) b e +5 \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) c d}{\sqrt {e}}}{2 c^{2}}\) | \(144\) |
| risch | \(\frac {x \sqrt {e \,x^{2}+d}}{2 c}-\frac {\frac {\left (2 b e -5 c d \right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{c \sqrt {e}}-\frac {\left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}-\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )^{2} e -\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}-\frac {b e -2 c d}{c}}}{x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}}\right )}{\sqrt {-\left (b e -c d \right ) e c}\, c \sqrt {-\frac {b e -2 c d}{c}}}+\frac {\left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}+\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )^{2} e +\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}-\frac {b e -2 c d}{c}}}{x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}}\right )}{\sqrt {-\left (b e -c d \right ) e c}\, c \sqrt {-\frac {b e -2 c d}{c}}}}{2 c}\) | \(540\) |
| default | \(\text {Expression too large to display}\) | \(3792\) |
-1/2/c^2*(-2*(b*e-2*c*d)^2/(e*(b*e-2*c*d)*(b*e-c*d))^(1/2)*arctanh((b*e-c* d)*(e*x^2+d)^(1/2)/x/(e*(b*e-2*c*d)*(b*e-c*d))^(1/2))-((e*x^2+d)^(1/2)*c*x *e^(1/2)-2*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))*b*e+5*arctanh((e*x^2+d)^(1/2 )/x/e^(1/2))*c*d)/e^(1/2))
Time = 0.54 (sec) , antiderivative size = 1079, normalized size of antiderivative = 7.76 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\left [\frac {2 \, \sqrt {e x^{2} + d} c e x - {\left (5 \, c d - 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (2 \, c d e - b e^{2}\right )} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} + 4 \, {\left ({\left (3 \, c^{2} d^{2} e^{2} - 5 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, c^{2} e}, \frac {2 \, \sqrt {e x^{2} + d} c e x - 2 \, {\left (5 \, c d - 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (2 \, c d e - b e^{2}\right )} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} + 4 \, {\left ({\left (3 \, c^{2} d^{2} e^{2} - 5 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )} \sqrt {e x^{2} + d} \sqrt {\frac {2 \, c d - b e}{c d e - b e^{2}}}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, c^{2} e}, \frac {2 \, \sqrt {e x^{2} + d} c e x + 2 \, {\left (2 \, c d e - b e^{2}\right )} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}} \arctan \left (\frac {{\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}}}{2 \, {\left ({\left (2 \, c d e - b e^{2}\right )} x^{3} + {\left (2 \, c d^{2} - b d e\right )} x\right )}}\right ) - {\left (5 \, c d - 2 \, b e\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right )}{4 \, c^{2} e}, \frac {\sqrt {e x^{2} + d} c e x - {\left (5 \, c d - 2 \, b e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, c d e - b e^{2}\right )} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}} \arctan \left (\frac {{\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-\frac {2 \, c d - b e}{c d e - b e^{2}}}}{2 \, {\left ({\left (2 \, c d e - b e^{2}\right )} x^{3} + {\left (2 \, c d^{2} - b d e\right )} x\right )}}\right )}{2 \, c^{2} e}\right ] \]
[1/4*(2*sqrt(e*x^2 + d)*c*e*x - (5*c*d - 2*b*e)*sqrt(e)*log(-2*e*x^2 + 2*s qrt(e*x^2 + d)*sqrt(e)*x - d) - (2*c*d*e - b*e^2)*sqrt((2*c*d - b*e)/(c*d* e - b*e^2))*log((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 2 4*b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e ^3)*x^2 + 4*((3*c^2*d^2*e^2 - 5*b*c*d*e^3 + 2*b^2*e^4)*x^3 + (c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(e*x^2 + d)*sqrt((2*c*d - b*e)/(c*d*e - b*e^2)))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e ^2)*x^2)))/(c^2*e), 1/4*(2*sqrt(e*x^2 + d)*c*e*x - 2*(5*c*d - 2*b*e)*sqrt( -e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (2*c*d*e - b*e^2)*sqrt((2*c*d - b *e)/(c*d*e - b*e^2))*log((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^ 2*e^2 - 24*b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2 + 4*((3*c^2*d^2*e^2 - 5*b*c*d*e^3 + 2*b^2*e^4)*x^3 + (c^2 *d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(e*x^2 + d)*sqrt((2*c*d - b*e)/ (c*d*e - b*e^2)))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d* e - b*c*e^2)*x^2)))/(c^2*e), 1/4*(2*sqrt(e*x^2 + d)*c*e*x + 2*(2*c*d*e - b *e^2)*sqrt(-(2*c*d - b*e)/(c*d*e - b*e^2))*arctan(1/2*(c*d^2 - b*d*e + (3* c*d*e - 2*b*e^2)*x^2)*sqrt(e*x^2 + d)*sqrt(-(2*c*d - b*e)/(c*d*e - b*e^2)) /((2*c*d*e - b*e^2)*x^3 + (2*c*d^2 - b*d*e)*x)) - (5*c*d - 2*b*e)*sqrt(e)* log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d))/(c^2*e), 1/2*(sqrt(e*x^2 + d)*c*e*x - (5*c*d - 2*b*e)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)...
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e} \,d x } \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2} \,d x \]